Spacetime Origins from Zeta Function Geometry: Analytical Continuation as the Physical Mechanism of Dimensional Emergence

This paper establishes a geometric theory of spacetime origins based on Riemann zeta function analytic structure in the complex plane. We prove that fundamental spacetime properties emerge from these geometric characteristics, particularly through zero distributions and functional equations. We propose a revolutionary theoretical framework explaining spacetime origins through the analytical continuation mechanism of the Riemann zeta function. Starting from the divergent quantum vacuum state, we demonstrate how finite physical dimensions emerge through the first step of analytical continuation, subsequently constructing curvature structures and quantum properties through hierarchical levels. Our core innovations include: (1) establishing correspondence between vacuum divergences and zeta function singularities; (2) proving dimensional emergence through the first analytical continuation step; (3) constructing a hierarchical theory from dimensions to curvature; (4) revealing the fundamental role of information conservation law $\mathcal{I}_{total} = \mathcal{I}_+ + \mathcal{I}_- + \mathcal{I}_0 = 1$ in spacetime generation; (5) establishing profound connections with string theory critical dimensions. This framework not only provides a mathematical mechanism for spacetime origins but also predicts observable physical effects, including fine structure of vacuum fluctuations, dynamical mechanisms of dimensional compactification, and microscopic implementation of the holographic principle. We demonstrate that spacetime dimensions correspond to critical values where $\zeta(d/2-1)$ is finite or zero, explaining string theory's critical dimensions: bosonic strings (26D) via $\zeta(-12) = 0$ and superstrings (10D) via $\zeta(-4) = 0$. The multi-dimensional negative information compensation network, governed by the series $\mathcal{I}_- = \sum_{n=0}^{N} \zeta(-2n-1)$, provides the stabilization mechanism preventing divergent growth while enabling creative dimensional emergence.